<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="vectsp_1.html#V7">distributive</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="polynom7.html#NM2">ConstPoly</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>







<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:49"/><i><font color="Green">E21</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> &amp;  <a href="polynom2.html#K5">eval</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="polynom2.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom2.html#D4">POLYNOM2:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1"/>
(  <a href="polynom7.html#K3" target="_self">coefficient</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> or  <a href="polynom7.html#K3" target="_self">coefficient</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1"><b>case </b></a><a NAME="E1:49_1_1_1"/><i><font color="Green">E22</font></i>: 
 <a href="polynom7.html#K3" target="_self">coefficient</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:49_1_1_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1_1_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E3:49_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polynom7.html#E1:49_1_1_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E5:49_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="polynom7.html#E1:49_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<a NAME="E6:49_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="polynom7.html#E1:49_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="polynom7.html#E24"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<a NAME="E7:49_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="polynom7.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E8:49_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom7.html#E1:49_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font> as  <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/><a NAME="E4:49_1_1_1"/>
 <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="polynom2.html#T20">POLYNOM2:20</a></i>;<br/><a NAME="E5:49_1_1_1"/><b>then </b>
 <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="polynom2.html#T9">POLYNOM2:9</a></i>;<br/><a NAME="E6:49_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="polynom7.html#E2:49"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E7:49_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/><b>hence </b><a NAME="E8:49_1_1_1"/>
 <a href="polynom2.html#K5">eval</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom7.html#K3" target="_self">coefficient</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:49"><i><font color="Green">E21</font></i></a>, <a class="txt" href="polynom7.html#E1:49_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2"><b>case </b></a><a NAME="E1:49_1_1_2"/><i><font color="Green">E22</font></i>: 
 <a href="polynom7.html#K3" target="_self">coefficient</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:49_1_1_2"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:49_1_1_2_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:49_1_1_2_1"/><i><font color="Green">E25</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polynom7.html#E1:49_1_1_2_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E4:49_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="polynom7.html#E2:49_1_1_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:49_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="polynom7.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E6:49_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom7.html#E1:49_1_1_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:49_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:49_1_1_2_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:49_1_1_2_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polynom7.html#E1:49_1_1_2_2"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E4:49_1_1_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="polynom7.html#E1:49_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="polynom7.html#E2:49_1_1_2_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="polynom7.html#E24"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:49_1_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:49_1_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:49_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E6:49_1_1_2"/><i><font color="Green">E25</font></i>: 
 <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="polynom2.html#T20">POLYNOM2:20</a></i>;<br/><a NAME="E7:49_1_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="polynom7.html#E4:49_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/><a NAME="E8:49_1_1_2"/><b>then </b>
 <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E9:49_1_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T33">FINSEQ_3:33</a></i>;<br/><a NAME="E10:49_1_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E11:49_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:49_1_1_2_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:49_1_1_2_3"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="polynom7.html#E1:49_1_1_2_3"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E4:49_1_1_2_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="polynom7.html#E2:49_1_1_2_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:49_1_1_2_3"/><i><font color="Green">E32</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:49_1_1_2_3_1"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E3:49_1_1_2_3_1"/>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:49_1_1_2_3_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="polynom7.html#E1:49_1_1_2_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="polynom7.html#E2:49_1_1_2_3_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:49_1_1_2_3_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="polynom7.html#E4:49_1_1_2_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="polynom7.html#E5:49_1_1_2_3_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:49_1_1_2_3"/>
( <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="polynom7.html#E9:49_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polynom7.html#E1:49_1_1_2_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E7:49_1_1_2_3"/><b>then </b><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="polynom7.html#E9:49_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polynom7.html#E1:49_1_1_2_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
 <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="polynom7.html#E7:49_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="polynom7.html#E7:49_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polynom7.html#E7:49_1_1_2_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E9:49_1_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom7.html#E6:49_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="polynom7.html#E9:49_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polynom7.html#E1:49_1_1_2_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="polynom7.html#E5:49_1_1_2_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:49_1_1_2"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="txt" href="polynom7.html#E10:49_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E13:49_1_1_2"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:49_1_1_2"/><i><font color="Green">E31</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom7.html#E12:49_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E15:49_1_1_2"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="polynom7.html#E9:49_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E16:49_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom7.html#E14:49_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E17:49_1_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom7.html#E7:49_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E18:49_1_1_2"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom7.html#E9:49_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E19:49_1_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom7.html#E7:49_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E20:49_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E21:49_1_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom7.html#E9:49_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polynom7.html#E19:49_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>then </b><i><font color="Green">E35</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E9:49_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E7:49_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polynom7.html#E18:49_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
 <a href="polynom7.html#K3" target="_self">coefficient</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="polynom7.html#E24"><i><font color="Green">Lemma15</font></i></a></i>
;<br/> <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E2:49"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><b>then </b><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="polynom7.html#E14:49_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="polynom2.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E2:49"><i><font color="Green">E21</font></i></a>, <a class="txt" href="polynom7.html#E13:49_1_1_2"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K3" target="_self">coefficient</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3">eval</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E17:49_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom7.html#E22:49_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="polynom2.html#D3">POLYNOM2:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K3" target="_self">coefficient</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom2.html#T16">POLYNOM2:16</a></i>
<br/>.= 
 <a href="polynom7.html#K3" target="_self">coefficient</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/><a NAME="E25:49_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K3" target="_self">coefficient</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:49"><i><font color="Green">E21</font></i></a>, <a class="txt" href="polynom7.html#E13:49_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>hence </b><a NAME="E26:49_1_1_2"/>
 <a href="polynom2.html#K5">eval</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom7.html#K3" target="_self">coefficient</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:49"><i><font color="Green">E21</font></i></a>, <a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:49"/>
 <a href="polynom2.html#K5">eval</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom7.html#K3" target="_self">coefficient</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div>
